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Theorem hbimtg 24163
Description: A more general and closed form of hbim 1725. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
hbimtg  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )

Proof of Theorem hbimtg
StepHypRef Expression
1 hbntg 24162 . . . 4  |-  ( A. x ( ph  ->  A. x ch )  -> 
( -.  ch  ->  A. x  -.  ph )
)
2 pm2.21 100 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  th ) )
32alimi 1546 . . . 4  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  th )
)
41, 3syl6 29 . . 3  |-  ( A. x ( ph  ->  A. x ch )  -> 
( -.  ch  ->  A. x ( ph  ->  th ) ) )
54adantr 451 . 2  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  ( -.  ch  ->  A. x
( ph  ->  th )
) )
6 ax-1 5 . . . . 5  |-  ( th 
->  ( ph  ->  th )
)
76alimi 1546 . . . 4  |-  ( A. x th  ->  A. x
( ph  ->  th )
)
87imim2i 13 . . 3  |-  ( ( ps  ->  A. x th )  ->  ( ps 
->  A. x ( ph  ->  th ) ) )
98adantl 452 . 2  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  ( ps  ->  A. x ( ph  ->  th ) ) )
105, 9jad 154 1  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527
This theorem is referenced by:  hbimg  24166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360
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